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I am interested in the generalized Stokes theorem and its various generalizations. It is apparent to me that many theorems in vector analysis and certain theorems in complex analysis can be viewed as special cases of the generalized Stokes theorem.

I am aware that de Rham cohomology provides an extension of the Stokes theorem to higher dimensions. Additionally, the Atiyah-Singer index theorem seems to be a notable generalization of the Stokes theorem, establishing a profound connection between the topological properties of a manifold and the analytic properties of specific differential operators on that manifold.

My question is whether there exist further generalizations of the Atiyah-Singer index theorem or if there is a "roof" theorem that encompasses the most general versions of the Stokes theorem and its subsequent generalizations. I am curious to know if there is an ultimate framework that unifies these theorems or if the process of generalization is seemingly endless, with new connections and generalizations continuously emerging.

I would appreciate any insights, references, or discussions on the topic to deepen my understanding of the generalizations of the Stokes theorem and their potential unifying frameworks.

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    $\begingroup$ I don't think this quite works: Atiyah-Singer is about calculating global topological invariants using local geometric data, whereas Stokes' theorem is about reducing local calculations on a manifold to local calculations on its boundary. When you "plug in" the de Rham complex to the Atiyah-Singer theorem (even generalizations to manifolds with boundary), you get the Gauss-Bonnet theorem, which is arguably a deeper result than Stokes' theorem. $\endgroup$
    – Paul Siegel
    Commented May 18, 2023 at 13:34
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    $\begingroup$ That said, you can make a connection using Poincare duality. The Atiyah-Singer theorem is essentially a concrete realization of Poincare duality for K-theory, using a functional-analytic model of the K-theory spectrum. Likewise, for ordinary de Rham cohomology, Poincare duality says that the integral of a closed top degree form on a closed manifold is zero, and this is precisely Stokes' theorem for closed manifolds. $\endgroup$
    – Paul Siegel
    Commented May 18, 2023 at 13:42
  • $\begingroup$ Poincaré duality is a great example but again I wonder because it also falls under this reccuring theme: "Information inside is described with something outside." I am curious now what is the primal root of that theme in mathematics. For example, a lot of theorems can be proved in real analysis using the Triangle inequality and I see why. The triangle inequality is such a natural thing and it is embedded in the world around us. Is there some "natural" reason why than all these other theorems obey this "inside" - "outisde" relation property? $\endgroup$
    – User198
    Commented May 18, 2023 at 15:02
  • $\begingroup$ The index theorem of Freed–Lott for differential K-theory should probably be mentioned in this context. $\endgroup$
    – Dmitri Pavlov
    Commented May 18, 2023 at 17:41
  • $\begingroup$ I think a good way to express the "inside/outside" theme is "when you sum differences, things often cancel". In fact this is the core idea of the 1d version of Stokes' theorem (the FTC): you just write down $f(b) - f(a) = f(b) - f(x_n) + f(x_n) - f(x_{n-1} + ... + f(x_1) - f(x_0) + f(x_0) - f(a)$ and figure out how to take a limit. The higher dimensional Stokes' theorem is just this same idea, but you match up faces of $n$-simplices with the opposite orientation. And indeed, the reason the idea keeps showing up is that there is a nice simplicial model for the (co)homology of manifolds. $\endgroup$
    – Paul Siegel
    Commented May 18, 2023 at 19:22

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